We construct virtual fundamental classes for dg-manifolds whose tangent sheaves have cohomology only in degrees 0 and 1. This condition is analogous to the existence of a perfect obstruction theory in the approach of Behrend and Fantechi fInvent. Math 128 (1997) 45-88] or Li and Tian uJ. Amer. Math. Soc. 11 (1998) 119-174]. Our class is initially defined in K-theory as the class of the structure sheaf of the dg-manifold. We compare our construction with that of Behrend and Fantechi as well as with the original proposal of Kontsevich. We prove a Riemann-Roch type result for dg-manifolds which involves integration over the virtual class. We prove a localization theorem for our virtual classes. We also associate to any dg-manifold of our type a cobordism class of almost complex (smooth) manifolds. This supports the intuition that working with dg-manifolds is the correct algebra-geometric replacement of the analytic technique of"deforming to transversal intersection".
- Virtual class